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<p>We propose a spatial discretization of the fourth-order nonlinear DLSS equation on the circle. Our choice of discretization is motivated by a novel gradient flow formulation with respect to a metric that generalizes martingale transport. The discrete dynamics inherits this gradient flow structure, and in addition further properties, such as an alternative gradient flow formulation in the Wasserstein distance, contractivity in the Hellinger distance, and monotonicity of several Lypunov functionals. Our main result is the convergence in the limit of vanishing mesh size. The proof relies on a discrete version of a nonlinear functional inequality between integral expressions involving second order derivatives.</p>

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A structure preserving discretization for the Derrida-Lebowitz-Speer-Spohn equation based on diffusive transport

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月9日

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  • Daniel Matthes,
  • Eva-Maria Rott,
  • Giuseppe Savaré,
  • André Schlichting

摘要

We propose a spatial discretization of the fourth-order nonlinear DLSS equation on the circle. Our choice of discretization is motivated by a novel gradient flow formulation with respect to a metric that generalizes martingale transport. The discrete dynamics inherits this gradient flow structure, and in addition further properties, such as an alternative gradient flow formulation in the Wasserstein distance, contractivity in the Hellinger distance, and monotonicity of several Lypunov functionals. Our main result is the convergence in the limit of vanishing mesh size. The proof relies on a discrete version of a nonlinear functional inequality between integral expressions involving second order derivatives.